剩余格上的α-交软滤子

赋予剩余格作为参数集,提出了剩余格上的α-交软滤子的概念,给出一些刻画和软集交运算下的性质。进一步,剩余格上的α-交软同余关系和α-交软滤子的关系被研究。特别是,当X=α时,证明了SFil(L)(剩余格上交软滤子α-交软滤子的全体)和Scon(L)(剩余格上交软滤子α-交软同余关系的全体)是完备格同构的。最后,得到了剩余格上的α-交软滤子像与原像的性质。     

剩余格的模糊滤子理论

运用模糊集的方法和原理进一步深入研究剩余格的滤子问题.在剩余格中引入了模糊预线性滤子,模糊可除滤子和模糊Glivenko滤子三类新的模糊滤子概念,给出了它们的若干性质和等价刻画.系统讨论了这三类模糊滤子以及模糊正关联滤子,模糊Boolean滤子,模糊MV滤子和模糊正则滤子间的相互关系,证明了一个模糊滤子为模糊MV滤子当且仅当它既是模糊正则滤子又是模糊可除滤子的结论.     

每个元有上覆盖的紧生成格的结构

Dilwrorth与Crawleyl973年提出能否去掉上半模格条件来刻画元素的不可约完全交既分解问题以及能否去掉强原子格的条件刻画紧生成格结构的问题,本文首先证明了每个元有上覆盖的紧生成格L中任意元有不可约完全交既分解,从而肯定地回答了Dilworth与Crawley上述第一个问题.之后,在每个元有上覆盖的紧生成格中引入局部强模格与局部强分配格的概念,研究了局部强模格中独立集的特性以及局部强模格与局部分配格的结构,从而部分解决了Dilworth与Crawley上述第二个问题.     

偏序集上的滤子极大理想

在偏序集上引入并考察了滤子极大理想的概念,证明了相应的存在性定理。引入并考察了伪极大元和伪既约元的概念,利用图表的形式对连续格中各种类型的既约元和素元之间的关系进行了归纳总结,完善了文献《Continuous Lattices and Domains》(作者:G.Gierz,et al)中的一个图表的相关内容,填补了在分配的连续格情形该图表的一个未知内容,部分地回答了该文献中的一个问题。     

幂交半格上的交同余及其性质

给出了幂交半格上几个具体的交同余,证明了任意多个交同余的并是交同余,并指出幂交半格的交同余类是子交半格,介绍了幂交半格上由θ诱导的交同余具有保并性和保序性。     

完全分配格与点格

本文第一部分利用完备格上的上拓扑子基,给出完全分配格与点格的若干新刻划,并讨论其上的 Scott 拓扑与 Lawson 拓扑的基与子基的构造.第二部分讨论点格与代数格的关系,证明了 L 是点格当且仅当 L 为代数格且 L~(op)为完全 Heyting代数,并证明了代数偏序集范畴与点格范畴是等价的.     

分子格的既约度,分解度与分子格乘积的既约性

为探讨分子格的乘积的既约性和乘积的既约分解,文献[1]提出了分子格的既约度和分解度的概念,本文是[1]的继续,进一步给出了分子格的既约度与分解度的一些性质,证明了关于分子格乘积的全体主子格之集的基数的一个定理,以及得出了一族分子格的乘积是既约分子格的一个充要条件,此外,本文还证明了分子格范畴中乘积对上积的完全分配性是自然同构,改正了[1]中对这一结论的证明。     

Heyting代数的扩张滤子

运用泛代数与逻辑学的方法和原理对Heyting代数中滤子概念作进一步研究.在Heyting代数H中引入了滤子F关于H的子集A的扩张滤子概念并考察其性质.证明了一个滤子F关于H的所有子集的扩张滤子全体之集构成一个完备Heyting代数且构成一个Stone格.     

分子格上伪补的构造

本文给出了分子格上伪补的构造定理，并证明了当一个非平凡分子格上有伪补，其既约因子上并不一定存在伪补，但其准既约因子上一定存在伪补，且任一分子格上的伪补都可表为其准既约因子上伪补的直积．     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

关于所有子模是纯子模的平坦模

引进了一新模类-完全平坦模(每一个商模平坦).并得到了:令M是平坦左R-模,RM是完全平坦模当且仅当RM的所有子模是纯的当且仅当每一个右R-模A是M-平坦的.同时本文用完全平坦模刻画了V.N.正则环.     

交一致连续偏序集

In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are:(1) A uniform complete poset L is meet uniform continuous iff ↑(U ∩↓ x) is a uniform Scott set for each x ∈ L and each uniform Scott set U;(2) A uniform complete poset L is meet uniform continuous iff for each∨∨x∈ L and each uniform subset S, one has x ∧S ={x ∧ s | s ∈ S}. In particular, a complete lattice L is meet uniform continuous iff L is a complete Heyting algebra;(3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous;(4) A uniform complete poset L is meet uniform continuous if L1 obtained by adjoining a top element1 to L is a complete Heyting algebra;(5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.     

Strongly continuous posets and the local Scott topology

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SC-posets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LC-continuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T₀-space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order.     

关于fuzzy映射的完全广义混合强非线性变分包含

引入并研究了Hilbert空间中一类新的关于fuzzy映射的完全广义混合强非线性变分包含，利用极大单调映射的预解算子技巧构造迭代算法．并证明此变分包含的解的存在性及由迭代算法所生成的迭代序列的收敛性。所得结果改进并推广以往所得的相应结果。     

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be \({{\bf jCRL} \cap {\bf mCRL}}\) . We prove

${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$

where the marked inclusions are proper.

Let L be a DL. Then \({L \in {\bf mCRL}}\) iff L has a distinguishing set of complete, prime filters. Similarly, \({L \in {\bf jCRL}}\) iff L has a distinguishing set of completely prime filters, and \({L \in {\bf CRL}}\) iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

一类群的幂半群的半格分解

讨论了一类群的幂半群的半格分解,得到了最大半格分解的刻划,及有关滤子,完全素理想等一系列结果,作为特例,得到了Putcha.M.S.关于有限群的幂半群的最大半格分解的结论。     

Note on residually finite rings

This paper is on the subject of residually finite (= RF) modules and rings introduced by Varadarajan [93] and [98/99]. Specifically there are several theorems that simplify proofs and generalize some results of Varadarajan, namely.

Theorem 1. An RF right R-module is finitely bedded (= has finite essential socle iff M is finite.

Corollay. If T is a right RF woth just finitely many simple ringht R-modules, them R is fimite.

Theorem 2. A commutative ring R is residually finite iff every local ring R_m at a maximal ideal m is finite.     

When Summands of Completely Decomposable Modules Are Completely Decomposable

We examine when summands of completely decomposable modules over a domain R are again completely decomposable. We show that this is the case if R is an h-local Prüfer domain. If R is 1-dimensional Noetherian, then the problem reduces locally if almost all localizations are integrally closed. If R is 1-dimensional Noetherian and local, then the integral closure of R must have at most two maximal ideals.